國立臺灣大學工學院土木工程學系 博士論文

### Department of Civil Engineering College of Engineering National Taiwan University

### Doctoral Dissertation

鋼筋混凝土剪力連接梁之剪力行為與設計

Cyclic Shear Strength and Seismic Design of Reinforced Concrete Coupling Beams

林孝勇 Erwin Lim

指導教授：黃世建 博士 Advisor: Shyh-Jiann Hwang, Ph. D

中華民國 104 年 6 月 June 2015

**To My parents, Families, and Tina **

**~ Ad Maiorem Dei Gloriam ~ **

**ACKNOWLEDGEMENT **

All praise, honor, and glory to my Lord Jesus Christ who has showered me with lots of love, mercy, and blessings so that I can complete my PhD program.

I would like to acknowledge The Ministry of Science and Technology of Taiwan, National Center for Research on Earthquake Engineering of Taiwan, and National Taiwan University who have provided research funding and support during my study. I also would like to express my deepest gratitude to Prof. Shyh-Jiann Hwang, who has been my advisor, mentor, guru, and friend at the same time. Prof. Hwang’s attitude towards research, teaching, and service to society has set a very high role model for me in pursuing my career.

My research in coupling beams would not be possible without great efforts from 鄭志宏, 王亭惟, 張于軒, 蔡尚錡, and 林秉誼 as well as 曾建創 who have carried out the experimental tests. I really appreciate the friendship from all lab mates in Prof. Hwang’s group, especially 李異安, 邱聰智, 翁撲文 and 沈文成 for fruitful discussions related to coursework and research; as well as having introduced me the taste of Kaoliang.

I would like to thank my parents and my “big” families for having provided wonderful opportunity for me to study in NTU. Their unfailing supports and continuous prayers were, are, and will always be my source of strength. I also enjoy the companionship from members in Holy Family Catholic Church Youth Club, especially 徐森義神父, 林 育如, and 鐘保祿. They have walked together with me, helped me, and taught me the beauty of silence and prayers. Last, but not the least, I am also grateful to have Tina Phang during my study and in my life in Taiwan. I cannot thank more for her love, understanding, and sacrifice.

**祈禱福傳會 每日奉獻誦 **

仁慈的天父， 感謝祢恩賜這新的一天， 我把今天的一切祈禱、工作、 喜樂和痛 苦全獻給祢。 願這奉獻，聯合祢的聖子耶穌聖心， 為救贖世界， 時時刻刻在感 恩聖事中奉獻自己。 願在聖神內， 偕同教會之母—聖母瑪利亞， 為祢的聖愛作 證， 並為本月內 教宗託付給我們的意向祈禱。

**ABSTRACT **

Coupling beams in the coupled wall system are the “fuse” to limit the input earthquake
force into building system. They are also expected to subject to very large displacement
demand. So, it is important to preserve the shear strength at the Maximum Considered
Earthquake (MCE) level. The requirement of the use of diagonal reinforcement layout
by ACI 318-14 code design procedure has been proven effective to maintain shear
strength at the MCE level and to achieve good deformation capacity, especially for deep
coupling beams (clear span-to-depth ratio A_{n}*h*<2.0 ). However, the ACI 318-14
neglects the check for flexural strength developed at the Design Based Earthquake
(DBE) level, which might cause adverse effects to the system. The design procedure of
ACI 318-14 is also criticized for lacking of a proper consideration of force transfer
mechanism. It causes that for coupling beams with intermediate span-to-depth ratio
(2.0≤A_{n}*h*≤4.0), engineers are left with options of choosing either a diagonally
reinforced layout or a conventional ductile beam design, without being fully aware of
the consequences.

This study summarized the findings from a five-year experimental program of coupling beam specimens tested in National Taiwan University from 2010 through 2014 and identified that the major parameters influencing their seismic behavior is the shear strength at the MCE level. Following it, semi rational shear strength models and the corresponding design procedures were developed. The design procedures suggested that the flexural design of a coupling beam be the primary strength design at the DBE level.

At the MCE level, by adopting a capacity design concept, the design procedure aims to provide sufficient shear capacity to resist the plastic shear demand. This study also proposes a concept of coupling beams with partial amount of diagonal reinforcement or

is suitable for coupling beams with intermediate span-to-depth ratio (2≤A_{n}*h*≤4). The
use of partial amount of diagonal reinforcement (hybrid) layout would not only maintain
the shear strength at the MCE level, but also ease the constructability of the coupling
beams.

Keywords: coupling beams, seismic behavior, shear strength, seismic design.

**TABLE OF CONTENT **

ACKNOWLEDGEMENT………iii

ABSTRACT……….. v

LIST OF TABLES.………xiii

LIST OF FIGURES……….xv

CHAPTER I INTRODUCTION………...1

1.1 Background and motivation………...1

1.2 Research objectives………... 4

1.3 Organization of thesis………5

CHAPTER II LITERATURE REVIEW………...7

2.1 Experimental research of coupling beam……….. 7

2.2 ACI 318-14 (2014) strengths formulation………...11

2.2.1 Flexural strength formulation based on ACI 318-14………...11

2.2.2 Shear strength formulation based on ACI 318-14………...13

2.2.3 ACI 318 strength formulation for diagonally reinforced coupling beam17 2.3 ACI 318-14 design procedure of a coupling beam………..18

2.4 Softened strut-and-tie (SST) model formulation……….22

2.4.1 General approach……….22

2.4.2 Simplified approach……….24

CHAPTER III EXPERIMENTAL STUDY OF COUPLING BEAMS………27

3.1 Experimental program……….29

3.2 Benchmark specimens……….31

3.2.1 Test specimens………32

3.2.2 Test results of benchmark specimens………..34

3.3 Specimens with rhombic layout………...43

3.3.1 Test specimens……….43

3.3.2 Test results of specimens with rhombic layout………43

3.3.3 Discussion on effects of rhombic layout……… 44

3.4 Specimens with vertically distributed longitudinal bars………..46

3.4.1 Test specimens………..46

3.4.2 Test results of specimens with vertically distributed longitudinal bars...47

3.4.3 Discussion on effects of vertically distributed longitudinal bars……… 48

3.5 Specimen casted with steel fiber-reinforced concrete……….50

3.5.1 Test specimen………..50

3.5.2 Test results of a specimen casted with steel fiber-reinforced concrete…50 3.5.3 Discussion on effects of steel fiber-reinforced concrete………..51

3.6 Specimens with partial amount of diagonal reinforcement (hybrid layout)……52

3.6.1 Test specimens……….52

3.6.2 Test results of specs with partial amount of diagonal reinforcement…..54

3.6.3 Discussion on effects of partial amount of diagonal reinforcement……57

3.7 Specimens with high strength material………58

3.7.1 Test specimens……….58

3.7.2 Test results of specimens with high strength material……….58

3.7.3 Discussion on effects of high strength material………...59

3.8 Specimen with discontinuous diagonal bars………61

3.8.1 Test specimen………..61

3.8.2 Test results of a specimen with discontinuous diagonal bars…………..61

3.8.3 Discussion on effects of discontinuous diagonal bars……….61

3.9 Specimen tested under axial restrain………...62

3.9.1 Test specimen………..62

3.9.2 Test results of a specimen tested under axial restrain……….62

3.9.3 Discussion on effect of axial restraint……….63

3.10 Summary on major findings in the experimental program………63

3.11 Lessons learned and extension to the development of an analytical model…..67

CHAPTER IV ANALYTICAL FORMULATION FOR SHEAR STRENGTH….. 71

Part I: Static Behavior………73

4.1 Deep beam with bearing plate………74

4.2 Deep beam with column stub……….85

4.3 Lessons learned and extension to the development of analytical model for specimens under cyclic loading………. 92

Part II: Cyclic Behavior……….94

4.4 Basic criteria for development of a shear strength model………..94

4.5 Shear strength modeling for deep coupling beams (A*n* *h*≤2.0)………...98

4.5.1 Shear strength at low displacement demand (or DBE level)…………...98

4.5.2 Shear strength at high displacement demand (or MCE level)………...100

4.5.3 Experimental verifications and discussions………...102

**4.6 Shear strength modeling for coupling beams with **A_{n}*h*>2.0**………... 113 **

4.6.1 Shear strength at low displacement demand (or DBE level)………….115

4.6.2 Shear strength at high displacement demand (or MCE level)………...116

4.6.3 Experimental verifications and discussions……….. 116

4.7 Alternative force transfer mechanism for coupling beams with A_{n}*h*=2.0...121

4.8 Summary………..122

CHAPTER V DESIGN IMPLEMENTATION………..125

5.1 Critical remarks on ACI 318-14 design procedure of a reinforced concrete
**coupling beam………..129 **

5.2 Proposed design procedure for deep coupling beams (A_{n}*h*≤2.0)………… 131

5.3 Parametric analysis of proposed design for deep coupling beams………..… 135

5.3.1 Coupling beams with straight diagonal bars layout………...135

5.3.2 Coupling beams with bent diagonal bars layout………142

5.3.3 Summary on the parametric analysis of the design procedure for deep coupling beams………..144

5.4 Proposed design procedure for coupling beams with intermediate span-to-
depth ratio (A*n* *h*>2.0)……...………...147

5.5 Parametric analyses of the proposed design for coupling beams with intermediate span-to-depth ratio ……….…... 151

5.5.1 Coupling beams with straight diagonal bars layout………... 151

5.5.2 Coupling beams with bent diagonal bars layout………... 156

5.5.3 Summary on the parametric analyses of the design procedure for intermediate/slender coupling beams……….... 158

5.6 Justification of the proposed design procedure to selected test specimens…. 159
5.7 Alternative design procedure for coupling beams with A_{n}*h*=2.0…………162

5.8 Comparisons between the proposed and ACI 318-14 design procedure…….163

CHAPTER VI CONCLUSIONS 6.1 Summary of findings………167

6.1.1 General comments……….167

6.1.2 Reinforcement layout………167

6.1.3 Flexural strength………168

6.1.4 Shear strength………169

6.1.5 Design limitations………. 172

6.2 Design recommendations……… 172

6.3 Future study……….174

REFERENCES………..175 APPENDIXES

**LIST OF TABLES **

Table 3.1 Reference list for specimen’s ID re-numbering ………...182

Table 3.2 Illustrative design variables………...183

Table 3.3 Test parameters………..………...188

Table 3.4 Material properties ………190

Table 3.5 Test results……….192

Table 4.1 Experimental verification for deep beams with bearing plate……...…194

Table 4.2 Experimental verification for deep beams with column stub……...….197

Table 4.3 Database of deep coupling beam specimens (A_{n}*h*≤2.0)………198

Table 4.4 Calculation results for deep coupling beams (A_{n}*h*≤2.0)……….…..200

Table 4.5 Experimental verification for deep coupling beams (A_{n}*h*≤2.0)……202

Table 4.6 Database of intermediate and slender coupling beam specimens
(A_{n}*h*>2.0)………...204

Table 4.7 Calculation results for intermediate and slender coupling beams…….206

Table 4.8 Experimental verification for intermediate and slender coupling beams
(A_{n}*h*>2.0)………..208

Table 4.9 Calculation results for coupling beams with A_{n}*h*=2.0 using DBD
mechanism……….210

Table 4.10 Experimental verification for coupling beams with A_{n}*h*=2.0 using
DBD mechanism………...210

Table 4.11 Comparisons for coupling beams with A_{n}*h*=2.0under different macro
models...………211

**LIST OF FIGURES **

Figure 1.1 Forces in a coupled wall system………213

Figure 1.2 Plastic hinge mechanisms in a coupled wall system………. 213

Figure 1.3 Deformation of a coupled wall system……….214

Figure 1.4 Failure of coupling beams after earthquake at Anchorage, Alaska…. 215 Figure 1.5 Pioneer tests of coupling beams……… 215

Figure 2.1 Test specimens of Paulay (1969)……….. 216

Figure 2.2 Test specimens of Barney et al. (1980)………. 217

Figure 2.3 Details of specimens C5 and C6 (Barney et al. 1980)………...218

Figure 2.4 Load deformation curves of specimens C5 and C6 (Barney et al. 1980)………... 219

Figure 2.5 Details of specimens CB-1A and CB-2A (Tassios et al. 1996)……….220

Figure 2.6 Details of specimens P02 and P07 (Galano and Vignoli 2000)……….220

Figure 2.7 Load-deformation curves of specimens CB-1A and CB-2A (Tassios et al. 1996)………...221

Figure 2.8 Load-deformation curves of specimens P02 and P07 (Galano and Vignoli 2000)………221

Figure 2.9 Details of specimens C1, C3,and C4 (Barney et al. 1980)………222

Figure 2.10 Load-deformation curves of C1, C3 and C4 (Barney et al. 1980)……223

Figure 2.11 Detailing failure of diagonal bar at bending location (Barney et al. 1980)………...224

Figure 2.12 Illustration of rhombic layout (Tegos and Penelis 1988)………..224

Figure 2.13 Details of specimen P14 (Galano and Vignoli 2000)………225

Figure 2.14 Load-deformation curve of specimen XX11

(Tegos and Penelis 1988)………...225

Figure 2.15 Load-deformation curve of specimen P14 (Galano and Vignoli 2000) 226 Figure 2.16 Details of specimen FB33 and CB33F (Naish et al. 2013)………226

Figure 2.17 Load-deformation curves of specimens FB33 and CB33F (Naish et al.2013a)……… ………..…..227

Figure 2.18 Details of specimens C7 and C8 (Barney et al. 1980)………...228

Figure 2.19 Load-deformation curve of specimens C7 and C8 (Barney et al. 1980)………... 229

Figure 2.20 Test specimens of Canbolat et al. (2005)……….. 230

Figure 2.21 Load-deformation curves of Canbolat et al. (2005)……….. 230

Figure 2.22 Details of specimen CB33D (Naish et al., 2013)……….. 230

Figure 2.23 Navier’s principles for flexural analysis………....231

Figure 2.24 Solution procedure for flexural analysis………231

Figure 2.25 Nominal flexural strength (ACI 318-14)………...232

Figure 2.26 B- and D-regions due to force discontinuity………. 232

Figure 2.27 B- and D-regions due to geometry……… 232

Figure 2.28 Solution procedure for the ACI 318-14 strut-and-tie model…………. 233

Figure 2.29 Determination of strut efficiency factor……… 234

Figure 2.30 Macro model of ACI 318 STM (2014)………..234

Figure 2.31 Shear strength of diagonally reinforced coupling beam………235

Figure 2.32 ACI 318 design concept of a coupling beam (Moehle et al. 2011)…...235

Figure 2.33 ACI 318-14 design flowchart of a beam with conventional reinforcement layout satisfying special moment resisting frame criteria ……… 236 Figure 2.34 Illustration on conventionally reinforced beam layout

(Moehle et al. 2011)………...237

Figure 2.35 ACI 318-14 design flowchart of a coupling beam……….238

Figure 2.36 Illustration on diagonally reinforced beam layout (Moehle et al. 2011)………...239

Figure 2.37 Softened strut-and-tie model formulation (Hwang and Lee 1999, Hwang and Lee 2000, Hwang et al. 2000)……….240

Figure 3.1 Experimental setup………241

Figure 3.2 Benchmark specimens……….. 242

Figure 3.3 Load-deformation curves of benchmark specimens………. 244

Figure 3.4 Crack patterns of benchmark specimens………... 246

Figure 3.5 Illustrations on improved deformation capacity ………...248

Figure 3.6 Stiffness degradation of benchmark specimens……….249

Figure 3.7 Determination of effective stiffness (EI)_{eff}………250

Figure 3.8 Pinching mechanism………. 251

Figure 3.9 Cumulative energy absorption of benchmark specimens………. 252

Figure 3.10 Specimens with rhombic layout……… 254

Figure 3.11 Load-deformation curves for rhombic layout………255

Figure 3.12 Crack patterns of rhombic layout……….. 256

Figure 3.13 Inadequacy of tension tie in bent bar………257

Figure 3.14 Stiffness degradation of rhombic layout………. .258

Figure 3.15 Cumulative energy absorption of rhombic layout……… 259

Figure 3.16 Moment-curvature analysis of conventional vs vertically distributed bars (Wong et al. 1990)……… 260

Figure 3.17 Specimens with bond enhancement layout………261 Figure 3.18 Load-deformation curves of specimens with

vertically distributed bars……….. 262

Figure 3.19 Crack patterns of specimens with vertically distributed bars………... 263

Figure 3.20 Strain gage reading for longitudinal reinforcement………. 264

Figure 3.21 Discontinuity among sets of spiral reinforcement……….264

Figure 3.22 Stiffness degradation curves of specimens with vertically distributed bars………. 265

Figure 3.23 Cumulative energy absorption curves of specimens with vertically distributed bars……… .266

Figure 3.24 Specimens with steel fiber reinforced concrete……… 267

Figure 3.25 Load-deformation curves of CB30-8……….268

Figure 3.26 Crack patterns of CB30-8………..268

Figure 3.27 Stiffness degradation curves of CB30-8………269

Figure 3.28 Cumulative energy absorption curves of CB30-8………. 269

Figure 3.29 Specimens with partial amount of diagonal reinforcement………….. 270

Figure 3.30 Load-deflection curves of specimens with partial amount of diagonal reinforcement……… 272

Figure 3.31 Crack patterns of specimens with partial amount of diagonal reinforcement……….274

Figure 3.32 Stiffness degradation curves of specimens with partial amount of diagonal reinforcement………. 276

Figure 3.33 Cumulative energy dissipation curves of specimens with partial amount of diagonal reinforcement……… 278

Figure 3.34 Specimens with high strength materials………280

Figure 3.35 Load-deformation curves of specimens with high strength materials.. 281

Figure 3.36 Crack patterns of specimens with high strength materials……… 282

Figure 3.37 Stiffness degradation curves of specimens with

high strength materials……….. 283

Figure 3.38 Cumulative energy absorption curves of specimens with high strength materials……….284

Figure 3.39 Specimen with discontinuous diagonal bars (CB30-4)………. 285

Figure 3.40 Load-deformation curves of CB30-4……… 285

Figure 3.41 Crack patterns curves of CB30-4………. 286

Figure 3.42 Inadequacy of shear friction capacity………286

Figure 3.43 Specimens with axial restrained (CB30-19-1)………. 287

Figure 3.44 Load-deformation curves of CB30-19-1……….. 287

Figure 3.45 Crack patterns of CB30-19-1……… 288

Figure 3.46 Axial force……… 288

Figure 3.47 Stiffness degradation curves of CB30-19-1………. 289

Figure 3.48 Cumulative energy absorption curves of CB30-19-1………289

Figure 3.49 Effect of the amount of diagonal reinforcement on the deformation capacity of coupling beam specimens……….. 290

Figure 3.50 Effect of the amount of diagonal reinforcement on the energy absorption of coupling beam specimens……… 291

Figure 3.51 Ease of steel assemblage……….. 291

Figure 4.1 Illustration of possible failure modes of an RC member………..292

Figure 4.2 Discontinuities in deep beam and coupling beam………. 293

Figure 4.3 Macro model for ACI 318-14 and its exp. verification (bearing plate)………... 294

Figure 4.4 Parametric models and experimental verifications (bearing plate)….. 294

**Figure 4.5 ** Experimental verification of Analysis 5 (bearing plate)………... 295

**Figure 4.6 ** Experimental verification of Analysis 6 (bearing plate)……….. 295
**Figure 4.7 ** Experimental verification of Analysis 7 (bearing plate)………295
Figure 4.8 Comparison of the strut efficiency factor and

the softening coefficient……… 296
**Figure 4.9 ** Experimental verification of Analysis 8 (bearing plate)……….. 296
**Figure 4.10 Experimental verification of Analysis 9 (bearing plate)……….. 296 **
Figure 4.11 Experimental test of deep beams (Foster and Gilbert 1998)…………. 297
Figure 4.12 Experimental test of pier caps (O’ Malley 2011)………. 297
Figure 4.13 Analytical study for column replaced with steel plate

(O’ Malley 2011)……….. 298

Figure 4.14 Analytical study for column replaced with stub column

(O’ Malley 2011)……….. 298

Figure 4.15 Failure of test 1-S (O’ Malley 2011)………. 299
Figure 4.16 Typical macro model for deep beam with column stub………300
Figure 4.17 Macro model for ACI 318-14 and its exp. Verification (column stub) 300
Figure 4.18 Parametric models and experimental verifications (column stub)……301
**Figure 4.19 Experimental verification of Analysis 5 (column stub)……… 302 **
**Figure 4.20 Experimental verification of Analysis 6 (column stub)……… 302 **
**Figure 4.21 ** Experimental verification of Analysis 7 (column stub)……… 302
**Figure 4.22 Experimental verification of Analysis 8 (column stub)……… 303 **
**Figure 4.23 Experimental verification of Analysis 9 (column stub)……… 303 **
**Figure 4.24 Force transfer mechanism for deep coupling beam……….. 304 **
**Figure 4.25 ** Force transfer mechanism for intermediate/slender coupling beam…. 304
**Figure 4.26 Idealized points for shear strength evaluation………..305 **
**Figure 4.27 Components of shear strength……….. 305 **

**Figure 4.28 ** Strain gage measurement of stirrups for deep coupling beams……….306
**Figure 4.29 ** Strain gage measurement of stirrups for intermediate coupling beams 307
**Figure 4.30 Shear strength model for a deep coupling beam………308 **
**Figure 4.31 Strain gage reading for diagonal bars (CB10-1)………309 **
Figure 4.32 Typical reinforcement layout of a coupling beam……….309
Figure 4.33 Strength ratios for specimens failing in shear (A_{n}*h*≤2.0)…………. 310
Figure 4.34 Strength ratios for specimens failing in flexure or flexure shear

(A*n* *h*≤2.0)……….. 310
Figure 4.35 Comparison of strength ratios predicted using ACI 318-14 and flexural

strength for diagonally reinforced specimen (A_{n}*h*≤2.0)………….. 311
Figure 4.36 Verification of proposed model at high displacement demand……….312
Figure 4.37 Force transfer mechanism for coupling beam with A_{n}*h*>2.0………314
Figure 4.38 Inclination angles of major cracks……….315
Figure 4.39 Identification of a shear element………315
Figure 4.40 Macro model for coupling beam withA*n* *h*>2.0………. 316
Figure 4.41 Strain gage reading for diagonal bars (CB30-19)………. 317
Figure 4.42 Strength ratios for specimens with A*n* *h*>2.0………. 318
Figure 4.43 Comparison of strength ratios predicted using ACI 318-14 and flexural

strength for diagonally reinforced specimen (A_{n}*h*>2.0)………319
Figure 4.44 Verification of proposed model at high displacement demand……….320
Figure 4.45 Comparison between DBD and direct mechanism for coupling beams

with A_{n}*h*=2.0……….323

Figure 5.1 Study case of CB20-3………324 Figure 5.2 Proposed design procedure for deep coupling beams………325

Figure 5.3 Typical reinforcement layout for a diagonally

reinforced coupling beam………..327

Figure 5.4 Beam dimension used in parametric analyses……….. 327 Figure 5.5 Parametric analyses for deep coupling beams with straight diagonal bars

*under different ratio of tension reinforcement (b=0.8h,* *f** _{c}*′ =28

*MPa*)..328 Figure 5.6 Study of concrete contribution for deep coupling beams with straight

diagonal bars under different ratio of tension reinforcement
( *f**c*′ =28*MPa, b=0.8h)……….. 329 *
Figure 5.7 Parametric analyses for deep coupling beams with straight diagonal bars

*under different b/h ratio (*ρ*st* =2.0%, *f** _{c}*′ =28

*MPa*)………. 330 Figure 5.8 Parametric analyses for deep coupling beams with straight diagonal bars under different concrete compressive strengths (ρ

*st*

*=1.5%, b=0.8h)…331*Figure 5.9 Parametric analyses for deep coupling beams with straight diagonal bars

under different concrete compressive strengths (ρ*st** =2.5%, b=0.8h)…332 *
Figure 5.10 Study of concrete contribution for deep coupling beams with straight

diagonal bars with under different concrete compressive strengths (ρ*st*

*=1.5%, b=0.8h)………. 333 *
Figure 5.11 Parametric analyses for deep coupling beams with straight diagonal bars

under effect of applying strength reduction factor (ρ_{st}* =1.5%, b=0.8h, *

*c* 56

*f*′ = *MPa*)……….. ..334

Figure 5.12 Parametric analyses for deep coupling beams with straight diagonal bars
under effect of applying strength reduction factor (ρ*st** =2.5%, b=0.8h, *

*c* 56

*f*′ = *MPa*)……… 335

Figure 5.13 Parametric analyses for deep coupling beams with straight diagonal bars
under effect of applying strength reduction factor (ρ*st** =1.5%, b=0.8h, *

*c* 28

*f*′ = *MPa*)……….335

Figure 5.14 Parametric analyses for deep coupling beams with straight diagonal bars
under effect of applying strength reduction factor (ρ_{st}* =2.5%, b=0.8h, *

*c* 28

*f*′ = *MPa*)……….336

Figure 5.15 Parametric analyses for deep coupling beams with bent diagonal bars
*under different ratio of tension reinforcement (b=0.8h,* *f** _{c}*′ =28

*MPa*)..337 Figure 5.16 Parametric analyses for deep coupling beams with bent diagonal bars

under different concrete compressive strength (ρ*st** =1.5%, b=0.8h)…. 338 *
Figure 5.17 Parametric analyses for deep coupling beams with bent diagonal bars

under effect of applying strength reduction factor (ρ*st** =1.5%, b=0.8h, *

*c* 56

*f*′ = *MPa*)……….339

Figure 5.18 Parametric analyses for deep coupling beams with bent diagonal bars
under effect of applying strength reduction factor (ρ*st** =2.5%, b=0.8h, *

*c* 28

*f*′ = *MPa*)……….340

Figure 5.19 Parametric analyses for deep coupling beams with bent diagonal bars
under effect of bent (ρ*st** =1.5%, b=0.8h, * *f**c*′ =56*MPa*, φ*s* =1.0 )…… 341
Figure 5.20 Parametric analyses for deep coupling beams under effect of bent

diagonal bars (ρ*st** =2.5%, b=0.8h, * *f**c*′ =28*MPa*, φ*s* =1.0 )………….. 342
Figure 5.21 Parametric analyses for deep coupling beams under effect of bent

diagonal bars (ρ*st** =1.5%, b=0.8h, * *f**c*′ =56*MPa*, φ*s* =0.85 )………… 342
Figure 5.22 Parametric analyses for deep coupling beams under effect of bent

Figure 5.23 Proposed design procedure for intermediate/slender coupling beams.. 344 Figure 5.24 Parametric analyses for intermediate coupling beams with straight

diagonal bars under different ratio of tension reinforcement
*(b=0.8h,* *f** _{c}*′ =28

*MPa*)………346 Figure 5.25 Study of concrete contribution for intermediate coupling beams with

straight diagonal bars under different ratio of tension reinforcement
( *f** _{c}*′ =28

*MPa, b=0.8h)……….. 347*Figure 5.26 Parametric analyses for intermediate coupling beams with straight

*diagonal bars under different of b/h ratio (*ρ*st* =2.0%, *f** _{c}*′ =28

*MPa*)…348 Figure 5.27 Parametric analyses for intermediate coupling beams with straight

diagonal bars under different concrete compressive strength (ρ*st* =1.5%,
*b=0.8h)……….. 349 *

Figure 5.28 Parametric analyses for intermediate coupling beams with straight diagonal bars under different concrete compressive strength

(ρ_{st}* =2.5%, b=0.8h)………350 *
Figure 5.29 Study of concrete contribution for intermediate coupling beams with

straight diagonal bars under different concrete compressive strength
(ρ*st** =1.5%, b=0.8h)……….. 351 *
Figure 5.30 Parametric analyses for intermediate coupling beams with straight

diagonal bars under effect of applying strength reduction factor

(ρ*st** =1.5%, b=0.8h, * *f** _{c}*′ =56

*MPa*)………352 Figure 5.31 Parametric analyses for intermediate coupling beams with straight

diagonal bars under effect of applying strength reduction factor

(ρ*st** =2.5%, b=0.8h, * *f**c*′ =56*MPa*)……… 353

Figure 5.32 Parametric analyses for intermediate coupling beams with straight diagonal bars under effect of applying strength reduction factor

(ρ*st** =1.5%, b=0.8h, * *f** _{c}*′ =28

*MPa*)……….353 Figure 5.33 Parametric analyses for intermediate coupling beams with straight

diagonal bars under effect of applying strength reduction factor

(ρ*st** =2.5%, b=0.8h, * *f**c*′ =28*MPa*)……… 354
Figure 5.34 Parametric analyses for intermediate coupling beams with bent diagonal

bars under different ratio of tension reinforcement

*(b=0.8h,* *f** _{c}*′ =28

*MPa*)………355 Figure 5.35 Parametric analyses for intermediate coupling beams with bent diagonal

bars under different concrete compressive strength

(ρ*st** =1.5%, b=0.8h)……….. .356 *
Figure 5.36 Parametric analyses for intermediate coupling beams with bent diagonal

bars under effect of applying strength reduction factor (ρ*st** =1.5%, b=0.8h, *

*c* 56

*f*′ = *MPa*)……… 357

Figure 5.37 Parametric analyses for intermediate coupling beams with bent diagonal
bars under effect of applying strength reduction factor (ρ*st** =2.5%, b=0.8h, *

*c* 28

*f*′ = *MPa*)……… 358

Figure 5.38 Parametric analyses for intermediate coupling beams under effect of bent
diagonal bars (ρ*st** =1.5%, b=0.8h, * *f** _{c}*′ =56

*MPa*, φ

*s*=1.0 )……… 359 Figure 5.39 Parametric analyses for intermediate coupling beams under effect of bent

diagonal bars (ρ*st** =2.5%, b=0.8h, * *f** _{c}*′ =28

*MPa*, φ

*s*=1.0 )……… 360 Figure 5.40 Parametric analyses for intermediate coupling under effect of bent

ρ ′ = φ

Figure 5.41 Parametric analyses for intermediate coupling beams under effect of bent
diagonal bars (ρ*st** =2.5%, b=0.8h, * *f**c*′ =28*MPa*, φ*s*=0.85 )………….. 361
Figure 5.42 Verifications on selected test specimens A_{n}*h*=1.0 (φ* _{s}* =1.0)…….. 362
Figure 5.43 Verifications on selected test specimens A

_{n}*h*=2.0(φ

*=1.0)……… 362 Figure 5.44 Verifications on selected test specimens A*

_{s}

_{n}*h*=2.4(φ

*=1.0)……...363 Figure 5.45 Verifications on selected test specimens A*

_{s}

_{n}*h*=3.0(φ

*=1.0)……….363 Figure 5.46 Verifications on selected test specimens A*

_{s}

_{n}*h*=3.3(φ

*=1.0)……….364 Figure 5.47 Verifications on selected test specimens A*

_{s}

_{n}*h*=4.0(φ

*=1.0)……….364 Figure 5.48 Parametric analyses for coupling beams with A*

_{s}

_{n}*h*=2.0 using direct and

45-degree force transfer mechanism

( *f** _{c}*′ =28

*MPa, b=0.8h,*φ

*=0.85)………..365 Figure 5.49 Parametric analyses for shear strength contributed by concrete strut with*

_{s}*n* *h*=2.0

A using direct and 45-degree force transfer mechanism
( *f** _{c}*′ =28

*MPa, b=0.8h,*φ

*=0.85)………. 366 Figure 5.50 Parametric analyses for coupling beams with A*

_{s}

_{n}*h*=2.0 using direct and

45-degree force transfer mechanism (ρ* _{st}* =1.5%

*, b=0.8h,*φ

*=0.85).. 367 Figure 5.51 Parametric analyses for shear strength contributed by concrete strut with*

_{s}*n* *h*=2.0

A using direct and 45-degree force transfer mechanism
(ρ* _{st}* =1.5%

*, b=0.8h,*φ

*=0.85)……….368 Figure 5.52 Parametric analyses for coupling beams with A*

_{s}

_{n}*h*=2.0 using direct transfer mechanism and 45-degree (ρ

*=2.5%*

_{st}*, b=0.8h,*φ

*=0.85)…369 Figure 5.53 Verifications on selected test specimens A*

_{s}

_{n}*h*=2.0using alternative (45- degree) force transfer mechanism (φ

*=1.0)……… 370*

_{s}Figure 5.54 Comparison of the steel requirement between ACI 318-14 and proposed method for deep coupling beams with straight diagonal bars

( φ* _{s}* =0.85)………371

Figure 5.55 Comparison of the steel requirement between ACI 318-14 and proposed
method for intermediate coupling beams with straight diagonal bars
( φ* _{s}* =0.85)……….. .372
Figure 6.1 Concept of coupling beams with partial amount of diagonal

reinforcement or hybrid layout

( *f** _{c}*′ =28

*MPa*,ρ

*=2.0% , φ*

_{st}*=0.85)……… 373*

_{s}**CHAPTER 1** **INTRODUCTION **

**1.1 Background and motivation **

Structural wall system is often adopted for high-rise buildings located in regions with high seismicity. The main reason behind this option is because the structural wall system not only provides higher lateral strength, but also control on lateral deformation.

In the real practice, structural wall system is usually encountered with openings in order to accommodate architectural considerations and/or mechanical/electrical installations.

One popular way out is by introducing the coupled wall system, where two separate structural walls are linked together using the coupling beams. By adopting this system, the designer can have advantages from additional redundancies created by the yielding of coupling beams.

Figure 1.1 illustrates the resisting mechanism of coupled wall system under assumed
static lateral load. The total resisting mechanism comes not only from the overturning
*moment of each wall (M**prw*), but also from the coupled axial force acting on the wall
due to accumulated shear force of coupling beams along the structure height (Σ*V** _{prw}*× ).

*L*In this way, under the similar force demand, the dimensions of walls in the coupled wall system are smaller than those of structural wall system without opening. Ultimately, the foundation design for each wall would be also greatly reduced (Paulay and Priestley 1992).

Another beneficial effect of having coupled wall system is due to the additional plastic mechanism occurred at coupling beams. With a good detailing of coupling beams, the formation of plastic hinges occurs not only on the base of shear wall, but also at both ends of coupling beams as shown in Fig. 1.2. Therefore, more redundancies are

In this particular coupled wall system, the role of the coupling beams becomes significant because they are required to maintain the overall structural integrity under a very large deformation as illustrated by Subedi (1991) in Fig. 1.3. To the author’s knowledge, the research on the behaviour of coupling beam did not draw the attention of many researchers until the first research on deep coupling beam (short span-to-depth ratio) started by Paulay in 1964 (Park and Paulay 2006). The importance of this research was confirmed through the severe damage of deep coupling beams during the earthquake at Anchorage, Alaska as shown in Fig. 1.4 (Paulay 1969).

As many as 12 deep coupling beam (clear span-to-depth ratio smaller than 2) specimens with conventional beam layout were built and tested (Paulay 1969). The test results on conventionally reinforced deep coupling beams indicated that there were significant pinching effects on the hysteresis loops and failures in sliding shear regardless of abundant amount of vertical shear reinforcement (Fig. 1.5a). In order to enhance the seismic performances, Paulay and Binney (1974) used diagonal reinforcement consistent with the internal resisting moment (Fig. 1.5b). This arrangement successfully improved the hysteresis loop becoming more robust, which ultimately changed the failure behaviour into bar buckling at the very high deformation level.

The superior performance of coupling beams reinforced with diagonal bars was
reconfirmed through various experimental studies by Barney et al. (1980), Tegos and
Penelis (1988), and Tassios et al. (1996). The ACI 318 building code later adopted this
recommendation in 1999 (ACI 318-99) and was revised in 2008 (ACI 318-08) and still
in use in 2014 (ACI 318-14). The ACI 318-14 (2014) regulated that for deep coupling
beams with clear span-to-depth ratios less than 2.0 (A_{n}*h*<2.0) and acting shear stress
υ*u* larger than 0.33 *f**c*′(*MPa*) have to be detailed using two intersecting groups of

diagonal bars. In addition, these groups of diagonal bars or the entire beam section must be properly confined to avoid diagonal bar buckling.

However, the current design equation of ACI 318-14 lacks of a rational model to
illustrate the behavior of a coupling beam. So, for coupling beams with intermediate
clear span-to-depth ratio (2.0≤A*n* *h*≤4.0), the ACI 318-14 code gives flexibility to
the engineers, whether to adopt diagonal or conventional reinforcement layout. The
available test data for coupling beams with 2.0≤A*n* *h*≤4.0 indicated that in general,
coupling beams detailed using diagonal layout gives robust hysteretic loops (Barney et
al. 1980, Fortney et al. 2008, and Naish et al. 2013a). However, these diagonally placed
diagonal bars sometimes create construction difficulty in the job site.

Canbolat et al. (2005) introduced the use of high-performance fiber-reinforced cement composite (HPFRCC) to simplify the transverse reinforcement detailing of a coupling beam with diagonal bars. The use of HPFRCC material contributed to the shear capacity and confinement of the beam which led to a concept that only a partial amount of diagonal bars need to be provided (Lequesne et al. 2010). Moehle et al. (2011) and Moehle (2015) referred this partial amount of diagonal bar concept as hybrid layout as it combines both conventional and diagonal layouts. This concept of coupling beam reinforced with a partial amount of diagonal bars greatly relieved the construction difficulty, but unfortunately no quantitative method was provided.

This thesis aims primarily to develop a semi-rational model to predict the shear strength of a coupling beam. Using this model, a design methodology is proposed accordingly.

Using the proposed design methodology, it is expected that a coupling beam which is easily constructed at the job site and possesses good deformation capacity can be designed. It is of major concern to keep this strength model as simple as possible, yet

So, in order to provide a comprehensive understanding of the overall structural behavior of a coupling beam, a total of five year research project with totally 40 specimens was carried out in National Taiwan University (NTU) with the research grant and testing facilities provided by National Science Council (NSC) and National Center for Research on Earthquake Engineering (NCREE) in Taiwan.

This five year research project was carried out by five master degree students of NTU
(Cheng 2010, Wang 2011, Chang 2012, Tsai 2013, and Lin 2014). The author
participated closely in these five year research project as a designer and responsible
mainly for the analytical work. The experimental program focused on studying the
seismic behavior of coupling beams with various parameters, which involved: clear
span-to-depth ratio (A_{n}*h*), reinforcement layout, amount of diagonal bars, compressive
strength of concrete, yield strength of diagonal bar, and yield strength of stirrups. This
thesis also compiles the main findings of the five year experimental studies.

**1.2 Research objectives **

This research has objectives as follows:

1. To understand the effects of several parameters on the seismic behavior of coupling beams. The major involved parameters are: clear span-to-depth ratio, reinforcement layout, amount of longitudinal bars, amount of diagonal bars, compressive strength of concrete, yield strength of diagonal bar, and yield strength of stirrups

2. To develop a shear strength model of a coupling beam. Along with the shear strength model, we need to: (1) identify the force transfer mechanism of a coupling beam subjected to cyclic loading, (2) estimate the contribution of concrete to the shear strength, (3) evaluate the contribution of diagonal bars to the shear strength, and (4) roles of vertical and horizontal shear reinforcement.

3. A particular interest of this study is focused on the use of coupling beams with a partial amount of diagonal reinforcement (hybrid layout). Based on the shear strength model, a design methodology is recommended so that engineers can quantize an appropriate amount of diagonal reinforcement.

**1.3 Organization of thesis **

This thesis is divided into six chapters. Chapter 1 outlines the research background, motivation, and objectives for this research. A more detailed study of available experimental study of reinforced concrete coupling beams is described in Chapter 2. In addition, Chapter 2 also illustrates the available analytical model for strength prediction.

Chapter 3 discusses the major finding of the five year experimental study of coupling beam conducted in NTU. Complete test matrix along with the test parameters and test results is given, but in-depth discussions are provided to key specimens only. These key specimens are grouped based on their test objectives. At the end of each group, based on the test parameters, comparisons between the test specimens and benchmark specimens are given. The last section of Chapter 3 summarizes the major findings during this five year research project and their extensions in developing analytical models.

Deep beam and coupling beam elements are parts of structural elements subjected to very high shear. The most prominent difference between these two is the loading condition, where the first is subjected to monotonic loading while the latter is subjected to cyclic loading. Hence, before developing the shear strength model for a coupling beam, the first part of Chapter 4 deals with a parametric study on the shear strength model for a deep beam database to identify the most important parameters affecting the accuracy of a strength model. Later on, based on the test observation and the parametric study, an analytical model for shear strength prediction of a coupling beam is developed

in the second part of Chapter 4. The accuracy of the proposed model is then gauged with available database of coupling beam specimens.

Chapter 5 extends the analytical model into the design recommendation for engineers.

This design flowchart is used to determine the appropriate amount of diagonal bars when designing a coupling beam. A series of parametric study was also carried out to evaluate the proposed design procedures. Finally, Chapter 6 provides the major conclusions of these series of study.

**CHAPTER 2 **

**LITERATURE REVIEW **

The literature study is separated into three sections. The first section reviews the experimental work of coupling beams and its related issues. The second and third sections focus on the available analytical tools for strength prediction of a beam. The analytical tools consists not only the ones adopted in ACI 318-14 (2014) code provision, but also a softened strut-and-tie (SST) model. The softened strut-and-tie model is considered as a rational model satisfying force equilibrium, strain compatibility, and constitutive equations.

**2.1 Experimental research of coupling beam **

The first research of coupling beam, to the author’s knowledge, was conducted in the
University of Canterbury (Paulay 1969). As many as 12 specimens was detailed using
conventional beam layout with major test parameters being clear span-to-depth ratio
(A_{n}*h*) of 1.02, 1.29, and 2.0, amount of vertical shear reinforcement (0.41% - 2.52%),
and type of loading (monotonic or cyclic loading) as shown in Fig. 2.1. Test results
indicated that most of these beams were subjected to axial elongation, causing a reduced
moment arm and ineffectiveness of compression reinforcement. Also, most of these
specimens were unable to reach their theoretical ultimate moment capacity due to the
inadequacy of their shear capacity.

To solve this issue, Paulay and Binney (1974) published test results comparing coupling
beams with conventional layout (A_{n}*h*=1.02)and diagonal layout (A_{n}*h*=1.29)as
shown previously in Fig. 1.5. From the hysteretic loops, it was clear that the seismic
behavior of coupling beam with diagonal reinforcement was much better compared to

that with conventional layout. Similar results were also concluded by Barney et al.

(1980), Tassios et al. (1996), and Galano and Vignoli (2000) as discussed below.

In 1980, researchers in Portland Cement Association (PCA) conducted eight coupling
beam specimens (Barney et al. 1980) as shown in Fig. 2.2. The first six specimens (C1
to C6) had A_{n}*h*=2.50 and the last two had A_{n}*h*=5.0. In relation with the behavior of
beam with short clear span-to-depth ratio, they compared the hysteretic loops of
specimen C5 and C6 (Figs. 2.3 and 2.4). The comparisons showed that the seismic
behavior of coupling beam with diagonal layout (C6) was much better than that with
conventional layout (C5). Similar conclusions were also obtained by comparing
specimens CB-1A and CB-2A with A_{n}*h*=1.0 (Tassios et al. 1996) as shown in Fig.

2.5 and specimens P02 and P07, both with A_{n}*h*=1.50 (Galano and Vignoli 2000) as
show in Fig. 2.6. The hysteretic loops of CB-1A and CB-2A (Fig. 2.7) and P02 and P07
(Fig. 2.8) indicated that specimens with diagonal layout possessed better deformation
capacity and energy absorption.

Following the advantages of having diagonal reinforcement within a coupling beam with low span-to-depth ratio, several detailing alternatives were also proposed. Within the eight specimens tested by Barney et al. (1980), three specimens (C1, C3, and C4) were detailed by having diagonal bars only in the hinge region (Fig. 2.9). Their test results showed that their original intention of putting diagonal reinforcement within the hinge region to eliminate the sliding shear was ineffective as shown by the hysteretic loops shown in Fig. 2.10. The main reason for this unsatisfactory improvement was because of the detailing failure at the location of bar bending (Fig. 2.11). This kind of detailing was also criticized to have improved the complexity in the steel assemblage.

Tegos and Penelis (1988) introduced the rhombic layout (Fig. 2.12) and followed by Galano and Vignoli (2000) in Fig. 2.13. They reported a satisfactory seismic behavior of

coupling beam with rhombic layout as shown in the hysteretic loops (Figs 2.14 and 2.15).

Naish et al. (2013a) tested specimens FB33 and CB33F with A_{n}*h*=3.33 as shown in
Fig. 2.16. Their test results showed diagonally reinforced specimen (CB33F) performed
much better compared to traditionally reinforced specimen (FB33), but FB33 still met
the standard as shown in Fig. 2.17. However, the presence of diagonal bar affected
significantly the overall energy dissipation. It is also noteworthy that the acting shear
stress for FB33 was quite low (<^{0.33} *f MPa**c*′

### ( )

).The difference in the deformation capacity for a specimen with diagonal layout and a
specimen with traditional layout becomes less significant as the clear span-to-depth ratio
(A_{n}*h*) increases. Specimens C7 and C8 (Fig. 2.18) with A_{n}*h*=5.0 by Barney et al.

(1980) showed little improvement on the deformation capacity of a coupling beam detailed with diagonal layout as compared to beam detailed with conventional layout (Fig. 2.19).

Although the use of diagonal reinforcement layout has been proven to bring beneficial
effect on the overall seismic behavior of a coupling beam, but its constructability has
been a major concern. Harries et al. (2005) claimed that although diagonally reinforced
coupling beam provides good seismic behavior, but it was simply impossible to design a
practically constructible one when the acting shear stress approaches the maximum ACI
318 stress limit, i.e. 0.83 *f MPa** _{c}*′( ). Meanwhile, Canbolat et al. (2005) and Naish et al.

(2013a) were concern about the constructability of confinement reinforcement on the groups of diagonal bars.

As the advancement of material science, new material was developed to enhance the tensile capacity of concrete. Canbolat et al. (2005) introduced the use of high-

reinforcement detailing of a coupling beam with diagonal bars. This HPFRCC improved not only the tensile capacity of concrete which leads to improved deformability, but also provided confinement effect. They also suggested bending the diagonal bars once they are within the mass concrete block. Figure 2.20 shows specimen 1 (with ordinary concrete) and specimen 4 (with HPFRCC). The hysteretic loops of these two specimens showed comparable seismic behavior (Fig. 2.21).

The use of HPFRCC was further investigated by Lequesne et al. (2010). They claimed that due to the improvement of tensile capacity of concrete, hence the required amount of diagonal bars can be reduced to 30% - 40%. Although no quantification method was proposed, this concept leads to an alternative detailing for a coupling beam with a partial amount of diagonal reinforcement.

Since 2008, the ACI 318 committee introduced two alternatives for the confinement detailing. The first alternative was the one in which confinement was provided for the diagonal bar groups only to prevent bar buckling, and the second alternative is to provide confinement on the entire beam section. In 2009, researchers in University of California at Los Angeles (UCLA) conducted the first test to evaluate these two alternatives (Naish et al. 2013a). Specimen CB33D (Fig. 2.22) only confined the diagonal bar groups, while specimen CB33F (Fig. 2.16b) confined the entire beam section. They found that beams with confinement on the entire section (CB33F) had slightly better seismic behavior compared to those with confinement on the diagonal bar groups (CB33D) as illustrated in Fig. 2.17b.

**2.2 ACI 318-14 (2014) strengths formulation **

In the design process, engineers must provide an adequate strength to a coupling beam when resisting the acting force. In general, two types of strength are of concern when designing a coupling beam, i.e.: flexural strength and shear strength. This section mainly introduces the ACI 318-14 (2014) approaches in calculating the flexural strength and shear strength of a flexural member. The flexural strength theory in ACI 318-14 which adopted the Bernoulli’s strain compatibility and the shear strength theory which adopted incomplete truss analogy are usually applicable for beams other than deep beams. Especially for coupling beams with diagonal reinforcement, the ACI 318-14 code also provides a specific equation to estimate its strength.

For deep beams, the ACI 318-14 recommends the strut-and-tie model (STM) provided in Chapter 23. However, the ACI 318-14 STM derived following the lower bound theory only requires that the force equilibrium is satisfied and neglects the strain compatibility and the material’s stress-strain relationship.

2.2.1 Flexural strength formulation based on ACI 318-14

The flexural strength of a beam can be calculated based on the well-developed flexural analysis. This flexural analysis was developed based on the Navier’s three principles in which force equilibrium, strain compatibility, and constitutive laws of material are satisfied. In many cases, except for deep member, the strain compatibility can be assumed to follow Bernoulli’s plane section remains plane before and after bending.

Given a beam section (Fig. 2.23) and by using the flexural analysis, one can develop a
moment-curvature (*M* −φ) diagram based on the solution procedure illustrated in Fig.

2.24.

*A beam section with beam width b, beam depth h, effective beam depth d calculated *
*A*

distance *d ′*from extreme compression fiber to centroid of compression reinforcement
*A′**s* are given for a flexural analysis. For each calculation point in *M* −φdiagram, one
can firstly assume the extreme compression fiber of concrete ε* _{c}*and an arbitrary value

*for depth of compression zone c. Then, by assuming Bernoulli’s strain compatibility,*the strain in compression reinforcement ε

*′and tensile reinforcement ε can be found*

_{s}*using similar triangle principle. Next, by using a given constitutive law for concrete and mild steel, the previously calculated strains can be transformed into stresses (*

_{s}*f*

*,*

_{c}*f ′*

*, and*

_{s}*f*

_{s}*). At this point, by iterating the depth of the compression zone c, equilibrium in*the horizontal forces must be sought as given in Eq. (2.1):

=0

−

′ +

=

Σ*F* *C*_{c}*C*_{s}*T* (2.1)

where *C** _{c}* is the compressive force of concrete,

*C′*

*is the force of compression reinforcement, and*

_{s}*T is the force of tension reinforcement, each obtained by multiplying*the respective stress with the area. Finally, the moment and curvature can be calculated using Eqs. (2.2) and (2.3), respectively:

### (

*d*

*y*

### )

*C*

### (

*d*

*d*

### )

*C*

*M* = * _{c}* − +

*′ − ′ (2.2)*

_{s}*c* *c*
ε

φ= (2.3)

*where y is the centroid of concrete compressive force measured from the extreme *
compressive fiber of concrete.

However, when evaluating the flexural strength, one simply needs to calculate the
*nominal flexural strength M**n* as illustrated in Fig. 2.25. The ACI 318-14 specifies that
*the M**n* is calculated when the extreme compression fiber ε equals 0.003 and the * _{c}*
concrete compressive stress can be simplified using Whitney’s stress block. Using the
Whitney’s stress block, the average concrete stress is taken as 0.85

*f ′*

*with depth of*

_{c}*block a equals*β , where _{1}*c* 0.65≤β_{1} =0.85−

### (

*f*

*′−28*

_{c}### )

7×0.05, where*f ′ is in MPa. In*

_{c}*this way, the nominal moment capacity M*

*n*can be calculated using Eq. (2.4):

### (

*d*

*d*

### )

*c* *C*
*d*
*C*

*M*_{n}* _{c}* ⎟+

*′ − ′*

_{s}⎠

⎜ ⎞

⎝⎛ −

= 2

β1

(2.4)

2.2.2 Shear strength formulation based on ACI 318-14

In general, a structural member can be divided into two regions, i.e.: B-region and D-
region. A B-region is a region where the stress distribution is more uniform and the
assumption of Bernoulli’s principal can be applied. On the other hand, D-region is the
region where some stress concentration may exist. The D-region may be caused due to a
stress concentration (such as a point load) or abrupt change in geometry. In ACI 318-14,
*the D-region can be determined as one beam depth h from the source of discontinuity. *

Figure 2.26 illustrates the determination of B- and D-regions for a simply supported beam subjected to a point load at the midspan, while Fig. 2.27 illustrated the B- and D- regions for a beam with fixed ends subjected to double curvature bending.

Since the stress distribution between these two regions is different, their shear strength mechanisms are also different. In ACI 318-14, the shear strength of B-region can be estimated using the incomplete truss analogy, while, that of D-region can be estimated using Strut-and-Tie model (STM). Then, the shear strength of a beam can be theoretically defined as the minimum of these two strengths.

Shear strength of B-region based on ACI 318-14

The ACI 318 shear strength provision for beam in B-region adopts the incomplete truss
analogy. The incomplete truss analogy acknowledges the shear contributed by concrete
*V**c** and shear contributed by vertical shear reinforcement V**s* as given in Eq. (2.5):

*s*
*c*

*n* *V* *V*

*V* = + (2.5)

The shear strength contributed by concrete was determined empirically and can be calculated using Eq. (2.6):

1 ( )

*c* 6 *c*

*V* = *f MPa bd*′ (2.6)

Meanwhile, the shear strength contributed by vertical shear reinforcement is estimated
using truss analogy by assuming a 45^{o} crack as given in Eq. (2.7):

*s*
*d*
*f*

*V** _{s}* =

*A*

^{vt}*(2.7)*

^{y}*where A**vt** is the area of vertical shear reinforcement within spacing s, and s is the center-*
to-center spacing of vertical shear reinforcement along beam axis.

Shear strength of D-region based on ACI 318-14 Strut-and-Tie Model

The solution procedure of ACI 318 STM can be presented in a simple form, as shown in
Fig. 2.28. Once all of the beam dimensions, reinforcement detailing, material properties,
and testing parameters are known, one may develop any macro model of a strut-and-tie
that satisfies force equilibrium. Then, based on the built macro model, the shear strength
*of a deep beam (V**n*) is determined as being the smallest of the following: the strengths
*of the diagonal strut at the top (V**1**) and at the bottom (V**2*), strengths of the nodal zone at
*the top (V**3**) and at the bottom (V**4**), and yielding of tension ties (V**5*) as shown in Eq.

(2.8):

### {

1, 2, 3, 4, 5### }

min

V* _{n}* =

*V*

*V*

*V*

*V*

*V*(2.8)

The strengths of the struts at the top and the bottom are determined from Eq. (2.9) and (2.10), respectively:

### (

0.85β### )

sinθV_{1} = _{s}*f ′*_{c}*A*_{cs}_{,}* _{top}* (2.9)

### (

0.85β### )

sinθV2 = *s**f ′**c* *A**cs*,*bot* (2.10)

Where β* _{s}* is the strut efficiency factor,

*A*

_{cs}_{,}

*and*

_{top}*A*

_{cs}_{,}

*are strut area at top and bottom, and θ is the inclination angle of strut.*

_{bot}The strut efficiency factor β* _{s}*is calculated based on the amount of provided vertical and
horizontal shear reinforcements as given in Eq. (2.11) and shown in Fig. 2.29:

003 . 0 sin ≥

Σ _{i}

*i*
*si*

*bs*

*A* α (2.11)

If the provided vertical and shear reinforcement satisfies Eq. (2.11) and the concrete
compressive strength *f** _{c}*′<42

*MPa*, the β

*can be taken as 0.75 and if not satisfying the criteria, the β*

_{s}*is taken as 0.6. For other cases, the β*

_{s}*is only allowed to be 0.4.*

_{s}Meanwhile, the strut areas at top and at bottom as well as the inclination angle of strut are determined following the developed macro model.

*The strengths of the nodal zone at the upper part (V**3**) and the lower part (V**4*) are
determined using Eqs. (2.12) and (2.13):

### (

0.85β### )

sinθV_{3}= _{n}*f ′*_{c}*A*_{cs}_{,}* _{top}* (2.12)

### (

0.85β### )

sinθV_{4} = _{n}*f ′*_{c}*A*_{cs}_{,}* _{bot}* (2.13)

where the efficiency factor of the nodal zone β* _{n}* for the CCC node (top node) is 1.0 and
CCT node (bottom node) is 0.8, respectively.

*Finally, the strength of the tension tie (V**5*) is taken as the yielding strength of flexural
reinforcement as shown in Eq. (2.14):

θ tan

V5 =*A**st**f**y* (2.14)

*where A**st* is the total area of tension reinforcement.

Macro Model for ACI 318 STM

One of the challenges of using strut-and-tie model is how to choose an appropriate force

transferred from the loading point (actuator) to the support. One may use a direct STM, in which the load is transferred directly from the loading plate to the reaction plate or other truss models to consider additional load paths due to the presence of vertical stirrups. The simplest STM, also adopted in this paper, uses a direct transfer mechanism in which the force from the loading actuator is directly transferred to the support reaction (Fig. 2.28). Brown and Bayrak (2008) also concluded that the direct transfer mechanism might be considered as an appropriate mechanism, especially for deep beams with a shear span to depth ratio that is less than 2.

After determining the load path, one needs to define the macro model of the strut-and-
*tie, which includes the determination of the width of the horizontal strut w**s*, the width of
*the horizontal tie w**t*, and the inclination angle of the diagonal strut θ . The
aforementioned width of the horizontal strut represents the depth of the compression
zone at the constant bending moment region. According to Tjhin and Kuchma (2002),
the depth of the compression zone is taken as the plastic compression zone shown in Eq.

(2.15):

0.85

*st* *y*
*s*

*s c*

*w* *A f*

β *f b*

= ′ (2.15)

*where A**st**f**y* is the yield strength of the main flexural reinforcement. Additionally, the
width of the horizontal tie is calculated by following the recommendation in ACI 318
Appendix A as shown in Eq. (2.16):

0.85

*st* *y*
*t*

*n c*

*w* *A f*

β *f b*

= ′ **(2.16) **

where β in this case is the efficiency factor of the nodal zone taken at the lower part *n*

(CCT node).